Schismatic family
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Five limit
The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its monzo is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the wedgie. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768.
The 5-limit version of the temperament is a microtemperament, sometimes called Helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. 53edo is a possible tuning for schismatic, but you need 118edo if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.
POTE generator: ~3/2 = 701.736
Mapping generator: ~3
Map: [<1 0 15|, <0 1 -8|]
EDOs: {{#rreplace: 12 17 24 29 36 41 53 65 77 82 89 94 101 106 118 130 135 142 147 154 159 171 183 195 200 207 212 219 224 236 248 253 260 265 272 277 289 301 313 318 325 330 342 354 366 371 378 383 390 395 407 419 424 431 436 443 448 460 472 484 489 496 501 508 513 525 537 542 549 554 561 566 578 590 602 607 614 619 626 631 643 655 660 667 672 679 684 696 708 720 725 732 737 744 749 761 773 778 785 790 797 802 814 826 838 843 850 855 862 867 879 891 896 903 908 915 920 932 944 956 961 968 973 985 997 1009 1014 1021 1026 1033 1038 1050 1062 1067 1074 1079 1086 1091 1103 1115 1127 1132 1139 1144 1151 1156 1168 1180 1185 1192 1197 1204 1209 1221 1233 1245 1250 1257 1262 1269 1274 1286 1298 1303 1310 1315 1322 1327 1339 1351 1363 1368 1375 1380 1387 1392 1404 1416 1421 1428 1433 1440 1445 1457 1469 1481 1486 1493 1498 1505 1510 1522 1534 1539 1546 1551 1558 1563 1575 1587 1599 1604 1611 1616 1628 1640 1652 1657 1664 1669 1676 1681 1693 1705 1710 1717 1722 1729 1734 1746 1758 1770 1775 1782 1787 1794 1799 1811 1823 1828 1835 1840 1847 1852 1864 1876 1888 1893 1900 1905 1912 1917 1929 1941 1946 1953 1958 1965 1970 1982 1994 2006 2011 2018 2023 2030 2035 2047 2059 2064 2071 2076 2083 2088 2100 2112 2124 2129 2136 2141 2148 2153 2165 2177 2182 2189 2194 2206 2218 2230 2242 2247 2259 2271 2283 2295 2300 2312 2324 2336 2348 2353 2365 2377 2389 2401 2418 2430 2442 2454 2471 2483 2495 2536 2548 2589 2601 2654 2707|/(\d+)(\D+)?/|\1\2}}
Customized val EDOs: 3456bc, 4205bc, 4954bc, 5703bc
Seven limit children
The second comma of the normal comma list defines which 7-limit family member we are looking at. Adding |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four.
Garibaldi
Commas: 225/224, 3125/3087
7-limit minimax tuning:
7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>, |5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>]
Eigenmonzos: 2, 7/6
9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>, |5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>]
Eigenmonzos: 2, 9/7
POTE generator: ~3/2 = 702.085
Mapping generator: ~3
Map: [<1 0 15 25|, <0 1 -8 -14|]
Wedgie: <<1 -8 -14 -15 -25 -10||
EDOs: 12, 29, 41, 53, 94, 241c, 335cd, 576cd
Badness: 0.0216
11-limit
Commas: 225/224, 385/384, 2200/2187
Minimax tuning:
[|1 0 0 0 0>, |25/16 1/8 0 -1/16 0>, |5/2 -1 0 1/2 0>, |25/8 -7/4 0 7/8 0>, |47/16 23/8 0 -23/16 0>]
Eigenmonzos: 2, 9/7
POTE generator: ~3/2 = 702.157
Mapping generator: ~3
Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|]
Edos: 41, 53, 94, 135, 229c, 323c, 417ce
Badness: 0.0274
13-limit
Commas: 225/224, 275/273, 325/324, 385/384
POTE generator: ~3/2 = 702.113
Mapping generator: ~3
Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|]
EDOs: 12, 41, 53, 94, 429cdef, 523cdef
Badness: 0.0207
Cassandra
Commas: 100/99, 225/224, 245/242
POTE generator: ~3/2 = 702.321
Mapping generator: ~3
Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|]
EDOs: 12, 29, 41, 123c, 217ce, 258ce
Badness: 0.0236
13-limit Cassandra
Commas: 100/99, 105/104, 196/195, 245/242
POTE generator: ~3/2 = 702.559
Mapping generator: ~3
Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|]
EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf
Badness: 0.0207
Helenus
Commas: 99/98 176/175 3125/3087
POTE generator: ~3/2 = 701.725
Mapping generator: ~3
Map: [<1 0 15 25 51|, <0 1 -8 -14 -30|]
EDOs: 12, 53, 118d, 171de
Badness: 0.0356
13-limit Helenus
Commas: 99/98, 176/175, 275/273, 847/845
POTE generator: ~3/2 = 701.747
Mapping generator: ~3
Map: [<1 0 15 25 51 56|, <0 1 -8 -14 -30 -33|]
EDOs: 53, 118d, 171de
Badness: 0.0263
Hemigari
Commas: 121/120, 225/224, 3125/3087
POTE generator: ~63/55 = 248.918
Mapping generator: ~110/63
Map: [<1 0 15 25 9|, <0 2 -16 -28 -7|]
EDOs: 29, 53, 82e, 135e, 188ce
Badness: 0.0507
13-limit
Commas: 121/120, 169/168, 225/224, 275/273
POTE generator: ~15/13 = 248.918
Mapping generator: ~26/15
Map: [<1 0 15 25 9 14|, <0 2 -16 -28 -7 -13|]
EDOs: 29, 53, 82e, 135ef, 188cef
Badness: 27.464
Guiron
Commas: 1029/1024, 10976/10935
7+9 limit: [|1 0 0 0>, |15/8 0 -1/8 0>, |0 0 1 0>, |65/24 0 1/24 0>]
Eigenmonzos: 2, 5/4
POTE generator: ~8/7 = 233.930
Mapping generator: ~8/7
Map: [<1 1 7 3|, <0 3 -24 -1|]
Wedgie: <<3 -24 -1 -45 -10 65||
EDOs: 36, 41, 77, 118, 277d
Badness: 0.0475
11-limit
Commas: 385/384, 441/440, 10976/10935
[|1 0 0 0 0>, |15/8 0 -1/8 0 0>, |0 0 1 0 0>, |65/24 0 1/24 0 0>, |37/6 0 -7/6 0 0>]
Eigenmonzos: 2, 5/4
POTE generator: ~8/7 = 233.931
Mapping generator: ~8/7
Map: [<1 1 7 3 -2|, <0 3 -24 -1 28|]
Edos: 41, 77, 118, 159, 200, 277d
Badness: 0.0266
13-limit
Commas: 196/195, 352/351, 385/384, 729/728
POTE generator: ~8/7 = 233.890
Mapping generator: ~8/7
Map: [<1 1 7 3 -2 0|, <0 3 -24 -1 28 19|]
EDOs: 41, 77, 118
Badness: 0.0284
Pogo
Commas: 32805/32768, 118098/117649
POTE generator: ~9/7 = 433.901
Mapping generator: ~9/7
Map: [<2 1 22 2|, <0 3 -24 5|]
EDOs: 36, 94, 130, 171, 183, 224, 354
Badness: 0.0796
11-limit
Commas: 540/539, 4000/3993, 32805/32768
POTE generator: ~9/7 = 433.911
Mapping generator: ~9/7
Map: [<2 1 22 2 25|, <0 3 -24 5 -25|]
EDOs: 36, 94, 130, 224, 354, 578
Badness: 0.0319
13-limit
Commas: 540/539, 729/728, 4000/3993, 4225/4224
POTE generator: ~9/7 = 433.911
Map: [<2 1 22 2 25 -2|, <0 3 -24 5 -25 13|]
EDOs: 36, 94, 130, 224, 354, 578
Badness: 0.0175
Sanjaab
Commas: 225/224, 3125/3087, 1331/1323
POTE generator: ~11/10 = 165.974
Map: [<1 2 -1 -3 0|, <0 -3 24 42 25|]
EDOs: 29, 65d, 94, 441cde, 535cde, 629cde
Badness: 0.0580
13-limit
Commas: 225/224, 275/273, 847/845, 1331/1323
POTE generator: ~11/10 = 165.963
Map: [<1 2 -1 -3 0 -1|, <0 -3 24 42 25 34|]
EDOs: 29, 65d, 94
Badness: 0.0338
Squirrel
Commmas: 686/675, 32805/32768
POTE generator: ~160/147 = 166.140
Map: [<1 2 -1 1|, <0 -3 24 13|]
EDOs: 29, 36, 65
Badness: 0.1747
11-limit
Commas: 245/242, 686/675, 896/891
POTE generator: ~11/10 = 166.097
Map: [<1 2 -1 1 0|, <0 -3 24 13 25|]
EDOs: 29, 36, 65
Badness: 0.0683
13-limit
Commas: 91/90, 169/168, 245/242, 896/891
POTE generator: ~11/10 = 166.054
Map: [<1 2 -1 1 0 3|, <0 -3 24 13 25 5|]
EDOs: 29, 36, 65f, 94df, 159df
Badness: 0.0437
Schism
Commas: 64/63, 360/343
POTE generator: ~3/2 = 701.556
Mapping generator: ~3
Map: [<1 0 15 6|, <0 1 -8 -2|]
Wedgie: <<1 -8 -2 -15 -6 18||
EDOs: 12, 41d, 53d
Badness: 0.0566
11-limit
Commas: 45/44, 64/63, 99/98
POTE generator ~3/2 = 702.136
Mapping generator: ~3
Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]
EDOs: 12, 29de, 41de
Badness: 0.0375
Pontiac
Commas: 32805/32768, 4375/4374
7-limit minimax:
[|1 0 0 0>, |74/47 0 -1/47 1/47>, |113/47 0 8/47 -8/47>, |113/47 0 -39/47 39/47>]
Eigenmonzos: 2, 7/5
9-limit minimax:
[|1 0 0 0>, |3/2 1/5 -1/10 0>, |3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>]
Eigenmonzos: 2, 10/9
POTE generator: 701.757
Mapping generator: ~3
Map: [<1 0 15 -59|, <0 1 -8 39|]
Wedgie: <<1 -8 39 -15 59 113||
Edos: 53, 118, 171, 1079, 1250, 1421
Badness: 0.0141
Grackle
Commas: 126/125, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: 701.239
Mapping generator: ~3
Map: [<1 0 15 -44|, <0 1 -8 -26|]
Wedgie: <<1 -8 -26 -15 -44 -38||
Badness: 0.070
Bischismic
Commas: 3136/3125, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: 701.592
Mapping generator: ~3
Map: [<2 0 30 69|, <0 1 -8 -20|]
Wedgie: <<2 -16 -40 -30 -69 -48||
Badness: 0.0547
11-limit
Commas: 441/440, 3136/3125, 8019/8000
POTE generator: ~3/2 = 701.612
Mapping generator: ~3
Map: [<2 0 30 69 102|, <0 1 -8 -20 -30|]
EDOs: 12, 118, 130, 248
Badness: 0.0282
13-limit
Commas: 441/440, 729/728, 1001/1000, 3136/3125
POTE generator: ~3/2 = 701.590
Mapping generator: ~3
Map: [<2 0 30 69 102 -75|, <0 1 -8 -20 -30 26|]
EDOs: 12, 118, 130, 248, 378
Badness: 0.0287
Kleischismic
Commas: 32805/32768, 1500625/1492992
POTE generator: ~36/35 = 50.920
Mapping generator: ~35/24
Map: [<2 1 22 -15|, <0 2 -16 19|]
Wedgie: <<4 -32 38 -60 49 178||
EDOs: 24, 94, 118, 212, 330, 542d, 872cd
Badness: 0.1106
11-limit
Commas: 385/384, 9801/9800, 14641/14580
POTE generator: ~36/35 = 50.918
Mapping generator: ~16/11
Map: [<2 1 22 -15 8|, <0 2 -16 19 -1|]
EDOs: 24, 94, 118, 212, 330e, 542de
Badness: 0.0367
13-limit
Commas: 352/351, 385/384, 729/728, 1575/1573
POTE generator: ~36/35 = 50.938
Mapping generator: ~16/11
Map: [<2 1 22 -15 8 15|, <0 2 -16 19 -1 -7|]
EDOs: 24, 94, 118, 212f
Badness: 0.0376
Term
Commas: 32805/32768, 250047/250000
7-limit minimax
Eigenmonzos: 2, 6/5
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: ~3/2 = 701.742
Mapping generator: ~3
Map: [<3 0 45 94|, <0 1 -8 -18|]
Wedgie: <<3 -24 -54 -45 -94 -58||
Badness: 0.0200
Sesquiquartififths
Commas: 2401/2400, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: ~448/405 = 175.434
Mapping generator: ~448/405
Map: [<1 1 7 5|, <0 4 -32 -15|]
Wedgie: <<4 -32 -15 -60 -35 55||
Badness: 0.0112
Sesquart
Commas: 243/242, 441/440, 16384/16335
POTE generator: ~256/231 = 175.406
Mapping generator: ~256/231
Map: [<1 1 7 5 2|, <0 4 -32 -15 10|]
EDOs: 41, 89, 130
Badness: 0.0293
13-limit
Commas: 243/242, 364/363, 441/440, 105644/105625
POTE generator: ~256/231 = 175.409
Map: [<1 1 7 5 2 -2|, <0 4 -32 -15 10 39|]
EDOs: 41, 89, 130, 301e, 431e
Badness: 0.0224
Bisesqui
Commas: 2401/2400, 9801/9800, 32805/32768
POTE generator: ~448/405 = 175.435
Map: [<2 2 14 10 23|, <0 4 -32 -15 -55|]
EDOs: 130, 212, 342, 1156, 1498
Badness: 0.0170
Sextilififths
Commas: 32768/32805, 235298/234375
POTE generator: ~21/20 = 83.053
Mapping generator: ~21/20
Map: [<1 2 -1 -1|, <0 -6 48 55|]
EDOs: 29, 101, 130, 289, 419
Badness: 0.1088
11-limit
Commas: 441/440, 4000/3993, 235298/234375
POTE generator: ~21/20 = 83.049
Mapping generator: ~21/20
Map: [<1 2 -1 -1 0|, <0 -6 48 55 50|]
EDOs: 29, 130, 289
Badness: 0.0455
13-limit
Commas: 363/363, 441/440, 676/675, 10985/10976
POTE generator: ~21/20 = 83.049
Mapping generator: ~21/20
Map: [<1 2 -1 -1 0 1|, <0 -6 48 55 50 39|]
EDOs: 29, 130, 289
Badness: 0.0253
Tsaharuk
Commas: 32805/32768, 420175/419904
POTE generator: ~243/224 = 140.350
Mapping generator: ~243/224
Map: [<1 1 7 0|, <0 5 -40 24|]
Wedgie: <<5 -40 24 -75 24 168||
EDOs: 17, 26, 43, 60, 77, 94, 171
Badness: 0.0307
Quanharuk
Commas: 16875/16807, 32805/32768
POTE generator: ~56/45 = 380.355
Mapping generator: ~56/45
Map: [<1 0 15 12|, <0 5 -40 -29|]
Wedgie: <<5 -40 -29 -75 -60 45||
EDOs: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd
Badness: 0.0720
11-limit
Commas: 540/539, 1375/1372, 32805/32768
POTE generator: ~56/45 = 380.352
Mapping generator: ~56/45
Map: [<1 0 15 12 -7|, <0 5 -40 -29 33|]
EDOs: 41, 142, 183, 224, 631d, 855d, 1079d
Badness: 0.0315
13-limit
Commas: 540/539, 1375/1372, 4096/4095, 6656/6655
POTE generator: ~56/45 = 380.351
Map: [<1 0 15 12 -7 -15|, <0 5 -40 -29 33 59|]
EDOs: 41, 142, 183, 224, 631d, 855d
Badness: 0.0214
Octant
Commas: 32805/32768, 2259436291848/2251875390625
POTE generator: ~3/2 = 701.713
Mapping generator: ~3
Map: [<8 0 120 -117|, <0 1 -8 11|]
Wedgie: <<8 -64 88 -120 117 384||
EDOs: 24, 224, 472, 696, 1168
Badness: 0.1572
11-limit
Commas: 9801/9800, 32805/32768, 46656/46585
POTE generator: ~3/2 = 701.713
Mapping generator: ~3
Map: [<8 0 120 -117 15|, <0 1 -8 11 1|]
EDOs: 24, 224, 472, 696, 1168
Badness: 0.0448
13-limit
Commas: 729/728, 1575/1573, 2200/2197, 6656/6655
POTE generator: ~3/2 = 701.725
Mapping generator: ~3
Map: [<8 0 120 -117 15 93|, <0 1 -8 11 1 -5|]
EDOs: 24, 224, 472, 696
Badness: 0.0304